Duality for convolution on subclasses of analytic functions and weighted integral operators

Abstract

In this article, we investigate a class of analytic functions defined on the unit open disc U={z:∣z∣0\alpha \gt 0, 0≤β≤10\le \beta \le 1, 00{\rm{Re}}\left\{\frac{f^{\prime} \left(z)+\frac{1-\gamma }{\alpha \gamma }z{f}^{^{\prime\prime} }\left(z)-\beta }{1-\beta }\right\}\gt 0 holds. We find conditions on the numbers α,β\alpha ,\beta , and γ\gamma such that Pα(β,γ)⊆SP(λ){{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma )\subseteq SP\left(\lambda ), for λ∈(−π2,π2)\lambda \in \left(-\frac{\pi }{2},\frac{\pi }{2}), where SP(λ)SP\left(\lambda ) denotes the set of all λ\lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α,β,γ\alpha ,\beta ,\gamma and the real-valued function φ\varphi so that the integral operator Vφ(f){V}_{\varphi }(f) maps the set Pα(β,γ){{\mathcal{P}}}_{\alpha }\left(\beta ,\gamma ) into the set SP(λ)SP\left(\lambda ), provided φ\varphi is non-negative normalized function (∫01φ(t)dt=1)\left({\int }_{0}^{1}\varphi \left(t){\rm{d}}t=1) and Vφ(f)(z)=∫01φ(t)f(tz)tdt.{V}_{\varphi }(f)\left(z)=\underset{0}{\overset{1}{\int }}\varphi \left(t)\frac{f\left(tz)}{t}{\rm{d}}t.

Keywords

• convolution
• hadamard product
• duality principle
• λ-spirallike functions
• dual set
• gaussian hypergeometric function
• 05a15
• 11b68
• 26b10
• 33e20